On the graph connectivity and the variable sum exdeg index

Jianwei Du; Xiaoling Sun; Jianwei Du; Xiaoling Sun

doi:10.3934/math.2021037

AIMS Mathematics

2021, Volume 6, Issue 1: 607-622. doi: 10.3934/math.2021037

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Research article

On the graph connectivity and the variable sum exdeg index

Jianwei Du ^,,
Xiaoling Sun

School of Science, North University of China, Taiyuan 030051, China

Received: 27 July 2020 Accepted: 20 October 2020 Published: 23 October 2020
MSC : 05C07, 05C35, 92E10

Topological indices are important descriptors which can be used to characterize the structural properties of organic molecules from different aspects. The variable sum exdeg index $SEI_{a}(G)$ of a graph $G$ is defined as $\sum _{u\in V(G)}d_{G}(u)a^{d_{G}(u)}$, where $d_{G}(u)$ is the degree of vertex $u$ and $a$ is an arbitrary positive real number different from 1. In this paper, we obtain the extremal values of the variable sum exdeg indices (for $a>1$) in terms of the number of cut edges, or the number of cut vertices, or the vertex connectivity, or the edge connectivity of a graph. Furthermore, the corresponding extremal graphs are characterized.
- variable sum exdeg index,
- cut edge,
- cut vertex,
- vertex connectivity,
- edge connectivity
Citation: Jianwei Du, Xiaoling Sun. On the graph connectivity and the variable sum exdeg index[J]. AIMS Mathematics, 2021, 6(1): 607-622. doi: 10.3934/math.2021037

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Abstract

Topological indices are important descriptors which can be used to characterize the structural properties of organic molecules from different aspects. The variable sum exdeg index $SEI_{a}(G)$ of a graph $G$ is defined as $\sum _{u\in V(G)}d_{G}(u)a^{d_{G}(u)}$, where $d_{G}(u)$ is the degree of vertex $u$ and $a$ is an arbitrary positive real number different from 1. In this paper, we obtain the extremal values of the variable sum exdeg indices (for $a>1$) in terms of the number of cut edges, or the number of cut vertices, or the vertex connectivity, or the edge connectivity of a graph. Furthermore, the corresponding extremal graphs are characterized.

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